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The coloring problem is a well-researched topic and its complexity is known for several classes of graphs. However, the question of its complexity remains open for the class of antiprismatic graphs, which are the complement of prismatic graphs and one of the four remaining cases highlighted by Lozin and Malishev. In this article we focus on the equivalent question of the complexity of the clique cover problem in prismatic graphs. A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbor in T. A graph is co-bridge-free if it has no (C_4+2K_1) as induced subgraph. We give a polynomial time algorithm that solves the clique cover problem in co-bridge-free prismatic graphs. It relies on the structural description given by Chudnovsky and Seymour, and on later work of Preissmann, Robin and Trotignon. We show that co-bridge-free prismatic graphs have a bounded number of disjoint triangles and that implies that the algorithm presented by Preissmann et al. applies.

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed (dge 1), is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for (mathcal{H})-subgraph-free graphs where (mathcal{H}) is any finite set of graphs.

Constrained clustering problems generalize classical clustering formulations, e.g., (k)-median, (k)-means, by imposing additional constraints on the feasibility of a clustering. There has been significant recent progress in obtaining approximation algorithms for these problems, both in the metric and the Euclidean settings. However, the outlier version of these problems, where the solution is allowed to leave out m points from the clustering, is not well understood. In this work, we give a general framework for reducing the outlier version of a constrained (k)-median or (k)-means problem to the corresponding outlier-free version with only ((1+varepsilon ))-loss in the approximation ratio. The reduction is obtained by mapping the original instance of the problem to (f(k,m, varepsilon )) instances of the outlier-free version, where (f(k, m, varepsilon ) = left( frac{k+m}{varepsilon }right) ^{O(m)}). As specific applications, we get the following results:

• First FPT (in the parameters k and m) ((1+varepsilon ))-approximation algorithm for the outlier version of capacitated (k)-median and (k)-means in Euclidean spaces with hard capacities.

• First FPT (in the parameters k and m) ((3+varepsilon )) and ((9+varepsilon )) approximation algorithms for the outlier version of capacitated (k)-median and (k)-means, respectively, in general metric spaces with hard capacities.

• First FPT (in the parameters k and m) ((2-delta ))-approximation algorithm for the outlier version of the (k)-median problem under the Ulam metric.

Our work generalizes the results of Bhattacharya et al. and Agrawal et al. to a larger class of constrained clustering problems. Further, our reduction works for arbitrary metric spaces and so can extend clustering algorithms for outlier-free versions in both Euclidean and arbitrary metric spaces.

Given a graph G and an integer b, Bandwidth asks whether there exists a bijection (pi ) from V(G) to ({1, ldots , |V(G)|}) such that (max _{{u, v } in E(G)} | pi (u) - pi (v) | le b). This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the tree-depth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number. In this paper we make progress in understanding the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number (omega ), thus significantly strengthening the previously mentioned result for vertex cover number. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results develop and generalize some of the methods of argumentation of the previous results and narrow some of the complexity gaps.

In bi-criteria optimization problems, the goal is typically to compute the set of Pareto-optimal solutions. Many algorithms for these types of problems rely on efficient merging or combining of partial solutions and filtering of dominated solutions in the resulting sets. In this article, we consider the task of computing the Pareto sum of two given Pareto sets A, B of size n. The Pareto sum C contains all non-dominated points of the Minkowski sum (M = {a+b|a in A, bin B}). Since the Minkowski sum has a size of (n^2), but the Pareto sum C can be much smaller, the goal is to compute C without having to compute and store all of M. We present several new algorithms for efficient Pareto sum computation, including an output-sensitive successive algorithm with a running time of (mathcal {O}(n log n + nk)) and a space consumption of (mathcal {O}(n+k)) for (k=|C|). If the elements of C are streamed, the space consumption reduces to (mathcal {O}(n)). For output sizes (k ge 2n), we prove a conditional lower bound for Pareto sum computation, which excludes running times in (mathcal {O}(n^{2-delta })) for (delta > 0) unless the (min,+)-convolution hardness conjecture fails. The successive algorithm matches this lower bound for (k in Theta (n)). However, for (k in Theta (n^2)), the successive algorithm exhibits a cubic running time. But we also present an algorithm with an output-sensitive space consumption and a running time of (mathcal {O}(n^2 log n)), which matches the lower bound up to a logarithmic factor even for large k. Furthermore, we describe suitable engineering techniques to improve the practical running times of our algorithms. Finally, we provide an extensive comparative experimental study on generated and real-world data. As a showcase application, we consider preprocessing-based bi-criteria route planning in road networks. Pareto sum computation is the bottleneck task in the preprocessing phase and in the query phase. We show that using our algorithms with an output-sensitive space consumption allows to tackle larger instances and reduces the preprocessing and query time compared to algorithms that fully store M.

The usual approach in runtime analysis is to derive estimates on the number of fitness function evaluations required by a method until a suitable element of the search space is found. One justification for this is that in real applications, fitness evaluation often contributes the most computational effort. A tacit assumption in this approach is that this effort is uniform and static across the search space. However, this assumption often does not hold in practice: some candidates may be far more expensive to evaluate than others. This might occur, for example, when fitness evaluation requires running a simulation or training a machine learning model. Despite the availability of a wide range of benchmark functions coupled with various runtime performance guarantees, the runtime analysis community currently lacks a solid perspective of handling variable fitness cost. Our goal with this paper is to argue for incorporating this perspective into our theoretical toolbox. We introduce two models of handling variable cost: a simple non-adaptive model together with a more general adaptive model. We prove cost bounds in these scenarios and discuss the implications for taking into account costly regions in the search space.

Spanner construction is a well-studied problem and Delaunay triangulations are among the most popular spanners. Tight bounds are known if the Delaunay triangulation is constructed using an equilateral triangle, a square, or a regular hexagon. However, all other shapes have remained elusive. In this paper, we extend the restricted class of spanners for which tight bounds are known. We prove that Delaunay triangulations constructed using rectangles with aspect ratio (A) have spanning ratio at most (sqrt{2} sqrt{1+A^2 + Asqrt{A^2 + 1}}), which matches the known lower bound.

An arborescence in a digraph is an acyclic arc subset in which every vertex except a root has exactly one incoming arc. In this paper, we show the reconfigurability of the union of k arborescences for fixed k in the following sense: for any pair of arc subsets that can be partitioned into k arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into k arborescences. This generalizes the result by Ito et al. (2023), who showed the case with (k=1). Since the union of k arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

The 2-opt heuristic is a simple local search heuristic for the travelling salesperson problem (TSP). Although it usually performs well in practice, its worst-case running time is exponential in the number of cities. Attempts to reconcile this difference between practice and theory have used smoothed analysis, in which adversarial instances are perturbed probabilistically. We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the perturbations are Gaussian. This model was previously used by Manthey and Veenstra, who obtained smoothed complexity bounds polynomial in n, the dimension d, and the perturbation strength (sigma ^{-1}). However, their analysis only works for (d ge 4). The only previous analysis for (d le 3) was performed by Englert, Röglin and Vöcking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in n and (sigma ^{-d}), and super-exponential in d. As the fact that no direct analysis exists for Gaussian perturbations that yields polynomial bounds for all d is somewhat unsatisfactory, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds for Euclidean 2-opt with Gaussian perturbations.